146 Frequently Asked Questions In Quantitative Finance
What is Meant by ‘‘Complete’’ and
‘‘Incomplete’’ Markets?
Short Answer
A complete market is one in which a derivative product
can be artificially made from more basic instruments,
such as cash and the underlying asset. This usually
involves dynamically rebalancing a portfolio of the
simpler instruments, according to some formula or algo-
rithm, to replicate the more complicated product, the
derivative. Obviously, an incomplete market is one in
which you can’t replicate the option with simpler instru-
ments.
Example
The classic example is replicating an equity option, a
call, say, by continuously buying or selling the equity so
that you always hold the amount
=e−D(T−t)N(d 1 ),
in the stock, where
N(x)=
1
√
2 π
∫x
−∞
e
−^12 φ^2
dφ
and
d 1 =
ln(S/E)+(r−D+^12 σ^2 )(T−t)
σ
√
T−t
.
Long Answer
A slightly more mathematical, yet still quite easily
understood, description is to say that a complete mar-
ket is one for which there exist the same number of
linearly independent securities as there are states of the
world in the future.
